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    Integrate e^-at(Si(t)-pi/2) from zero to infinity

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    Evaluate the integral with boundaries 0 to infinity:
    J(a) = e^-at(Si(t)-pi/2)dt.

    © BrainMass Inc. brainmass.com March 5, 2021, 12:44 am ad1c9bdddf
    https://brainmass.com/math/integrals/integrate-infinity-530966

    Solution Preview

    The integral:

    J(a) = Integral from 0 to infinity of exp(-at)[Si(t) -pi/2] dt

    can be evaluated as follows. Inserting the expression Si(t) = integral from 0 to t of sin(x)/x dx gives:

    J(a) = Integral over t from 0 to infinity of exp(-at)[Integral from 0 to t of sin(x)/x dx -pi/2] dt

    Since Integral from 0 to infinity of sin(x)/x dx = pi/2, we can write this as:

    J(a) =- Integral over t from 0 to infinity of exp(-at) Integral from x = t to infinity of sin(x)/x dx dt =

    =- Integral over t ...

    Solution Summary

    We explain in detail how the integral can be computed.

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